With the chiral dimensions in hand, it is now possible to restrict the central charges of the putative 4d \({\cal N}=2\)  theory to a discrete set of possibilities. Indeed, the following standard formulae can be used [1]}, [2]}, [3]},To be precise, in the twisted case the \(h_*\)  appearing in (REF ) is one of the chiral dimensions of the \(S\) -tranformed VOA—see e.g. [3]}. Here we are focussing on the untwisted case.
\(c_{{\mathrm {4d}}} = -{ c_{{\mathrm {2d}}} \over 12}~, \hspace{36.135pt} a_{\mathrm {4d}} = {1\over 48}(24 h_* - 5 c_{\mathrm {2d}})~.\) 
